This problem involves calculating the perimeter of a rectangle based on its relationship with a square's area and perimeter. Let's break it down step-by-step.

Finding the Square's Dimensions

We are given that the perimeter of the square is 56 m. The formula for the perimeter of a square is:

$$ \text{Perimeter}_{\text{square}} = 4 \times \text{side} $$

Using the given information:

$$ 56 \text{ m} = 4 \times \text{side} $$

To find the length of one side of the square, we divide the perimeter by 4:

$$ \text{side} = \frac{56 \text{ m}}{4} = 14 \text{ m} $$

Calculating the Square's Area

Now that we have the side length of the square, we can calculate its area. The formula for the area of a square is:

$$ \text{Area}_{\text{square}} = \text{side}^2 $$

Substituting the side length we found:

$$ \text{Area}_{\text{square}} = (14 \text{ m})^2 = 196 \text{ m}^2 $$

Determining the Rectangle's Area

The problem states that the area of the square is double the area of the rectangle. So:

$$ \text{Area}_{\text{square}} = 2 \times \text{Area}_{\text{rectangle}} $$

We can rearrange this to find the rectangle's area:

$$ \text{Area}_{\text{rectangle}} = \frac{\text{Area}_{\text{square}}}{2} $$

Plugging in the square's area:

$$ \text{Area}_{\text{rectangle}} = \frac{196 \text{ m}^2}{2} = 98 \text{ m}^2 $$

Finding the Rectangle's Dimensions

Let the breadth of the rectangle be '$b$' and the length be '$l$'. We are given that the length of the rectangle is double its breadth:

$$ l = 2b $$

The formula for the area of a rectangle is:

$$ \text{Area}_{\text{rectangle}} = l \times b $$

Substitute '$l = 2b$' into the area formula:

$$ \text{Area}_{\text{rectangle}} = (2b) \times b = 2b^2 $$

We know the area of the rectangle is 98 m², so:

$$ 98 \text{ m}^2 = 2b^2 $$

Now, solve for '$b^2$':

$$ b^2 = \frac{98 \text{ m}^2}{2} = 49 \text{ m}^2 $$

Take the square root to find the breadth '$b$':

$$ b = \sqrt{49 \text{ m}^2} = 7 \text{ m} $$

Now find the length '$l$' using the relation $l = 2b$:

$$ l = 2 \times 7 \text{ m} = 14 \text{ m} $$

Calculating the Rectangle's Perimeter

Finally, we calculate the perimeter of the rectangle using its length and breadth. The formula for the perimeter of a rectangle is:

$$ \text{Perimeter}_{\text{rectangle}} = 2(l + b) $$

Substitute the values of '$l$' and '$b$':

$$ \text{Perimeter}_{\text{rectangle}} = 2(14 \text{ m} + 7 \text{ m}) $$

$$ \text{Perimeter}_{\text{rectangle}} = 2(21 \text{ m}) $$

$$ \text{Perimeter}_{\text{rectangle}} = 42 \text{ m} $$

Conclusion

Therefore, the perimeter of the rectangle is 42 m.